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Unformatted text preview: Chapter 3 Linear Algebra 3.1 Fields Consider a set F of objects and two operations on the elements of F , addition and multiplication. For every pair of elements s,t F then ( s + t ) F . For every pair of elements s,t F then st F . Suppose that these two operations satisfy 1. addition is commutative: s + t = t + s s,t F 2. addition is associative: r + ( s + t ) = ( r + s ) + t r,s,t F 3. to each s F there exists a unique element ( s ) F such that s +( s ) = 0 4. multiplication is commutative: st = ts s,t F 5. multiplication is associative: r ( st ) = ( rs ) t r,s,t F 6. there is a unique nonzero element 1 F such that s 1 = s s F 7. to each s F there exists a unique element s 1 F such that ss 1 = 1 8. multiplication distributes over addition: r ( s + t ) = rs + rt r,s,t F . Then, the set F together with these two operations is a feld . Example 3.1.1. The real numbers with the usual operations of addition and multi plication form a Feld. The complex numbers with these two operations also form a Feld. 34 3.2. MATRICES 35 Example 3.1.2. The set of integers with addition and multiplication is not a Feld. P 3.1.3. Is the set of rational numbers a subFeld of the real numbers? Example 3.1.4. Is the set of all real numbers of the form s + t 2 , where s and t are rational, a subFeld of the complex numbers? The set F = s + t 2 : s,t Q together with the standard addition and multiplication is a Feld. Let s,t,u,v Q , s + t 2 + u + v 2 = ( s + u ) + ( t + v ) 2 F s + t 2 u + v 2 = ( su + 2 tv ) + ( sv + tu ) 2 F s + t 2 1 = s t 2 s 2 + 2 t 2 = s s 2 + 2 t 2 t s 2 + 2 t 2 2 F Again, the remaining properties are straightforward to prove. The Feld s + t 2 , where s and t are rational, is a subFeld of the complex numbers. 3.2 Matrices Let F be a feld and consider the problem oF fnding n scalars x 1 ,...,x n which satisFy the conditions a 11 x 1 + a 12 x 2 + + a 1 n x n = y 1 a 21 x 1 + a 22 x 2 + + a 2 n x n = y 2 . . . . . . . . . . . . a m 1 x 1 + a m 2 x 2 + + a mn x n = y m (3.1) where { y i : 1 i n } F and { a ij : 1 i m, 1 j n } F . These conditions Form a system of m linear equations in n unknowns . A shorthand notation For (3.1) is the matrix equation Ax = y , where A is the matrix representation given by A = a 11 a 12 a 1 n a 21 a 22 a 2 n . . . . . . . . . . . . a m 1 a m 2 a mn 36 CHAPTER 3. LINEAR ALGEBRA and x , y denote x = ( x 1 ,...,x n ) T y = ( y 1 ,...,y m ) T . Defnition 3.2.1. Let A be an m n matrix over F and let B be an n p matrix over F . the matrix product AB is the m p matrix C whose i,j entry is c ij = n r =1 a ir b rj ....
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 Fall '08
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